Compactness conditions in the geometric theory of Banach spaces

Abstract

IN THE last few years a lot of papers have appeared devoted to the branch of geometric theory of Banach spaces involving compactness conditions. The notions being fundamental in the classical geometry of Banach spaces, such as strict convexity, uniform convexity, smoothness and uniform smoothness [ 141 have been translated in terms of a measure of noncompactness by Huff [13], Goebel and Sekowski [ 111, Sekowski and Stachura [25], Rolewicz [22, 231, Montesinos [18] and the present author [2, 31. These new concepts turn out to be useful tools both in the geometric theory of Banach spaces and for many applications (cf. [3, 6, 10, 11, 13, 18, 20, 21, 241). The purpose of this paper is to discuss several properties connected with the concepts mentioned above. Firstly, we define the modulus of near convexity which corresponds to the concept introduced by Huff [13]. It will be shown that this modulus is, in a certain sense, the inverse function to the modulus of noncompact convexity introduced by Goebel and Sekowski [ 111. Secondly, using the concept of a nearly uniformly smooth Banach space [3] we introduce the notion of a modulus of near smoothness and we describe a certain relationship between this modulus and the modulus of near convexity. This relationship is an analogue of the classical Lindenstrauss dual formula [ 161. Based on this relationship, and on properties of the mentioned moduli, we can calculate several formulas for the moduli of noncompact convexity, near convexity and near smoothness in some classical Banach spaces. We also extend the classical results due to Day [7, 81 concerning the classification of the product spaces. Namely, we shall show that some product spaces which are not uniformly convex are nearly uniformly convex or have the so-called drop property. Similar results concerning the concept of smoothness will be also obtained. The results of this paper form a continuation of the study from a previous article [3].